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We count $S$-integer inputs $a$ for which $f(a)$ has a $k$-rational preimage under $g$, after removing the polynomial graph components $Y=h(X)$ with $f=g\\circ h$. The main theorem gives componentwise height bounds. For a rational component of $f(X)-g(Y)=0$ with one geometric point at infinity and projection degree $d_X(C)$ to the $X$-line, the corresponding contribution has the sharp power-log order $B^{[k:\\mathbb{Q}]/d_X(C)}(\\log B)^{q_{k,S}}$, where $q_{k,S}=\\mathrm{rk}\\,\\mathcal{O}_{k,S}^{\\ast}=|S|-1$, precisely when its","authors_text":"Henry Shin","cross_cats":["math.AG"],"headline":"Rational components with one geometric point at infinity contribute sharp power-log order B^{[k:Q]/d_X(C)} (log B)^{|S|-1} when S-active.","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-13T02:28:01Z","title":"Componentwise height bounds for polynomial value-set lifting"},"references":{"count":17,"internal_anchors":0,"resolved_work":17,"sample":[{"cited_arxiv_id":"","doi":"10.1142/s1793042109002274","is_internal_anchor":false,"ref_index":1,"title":"P. Alvanos, Y. F. Bilu, and D. Poulakis,Characterizing algebraic curves with in- finitely many integral points, Int. J. Number Theory5(2009), no. 4, 585–590, DOI 10.1142/S1793042109002274","work_id":"d44aae11-5d19-4075-b675-5bb23a82033e","year":2009},{"cited_arxiv_id":"","doi":"10.4064/aa99-3-2","is_internal_anchor":false,"ref_index":2,"title":"R. M. Avanzi and U. M. Zannier,Genus one curves defined by separated variable polynomials and a polynomial Pell equation, Acta Arith.99(2001), no. 3, 227–256, DOI 10.4064/aa99-3-2","work_id":"64872be9-f81e-46a0-95f1-7531f4a31767","year":2001},{"cited_arxiv_id":"","doi":"10.4064/aa167-1-4","is_internal_anchor":false,"ref_index":3,"title":"Barroero,AlgebraicS-integers of fixed degree and bounded height, Acta Arith","work_id":"45039e86-9e5d-426d-b28d-87bc1d7ce3f7","year":2015},{"cited_arxiv_id":"","doi":"10.4064/aa-90-4-341-355","is_internal_anchor":false,"ref_index":4,"title":"Y. F. Bilu,Quadratic factors off(x)−g(y), Acta Arith.90(1999), no. 4, 341–355, DOI 10.4064/aa-90-4-341-355","work_id":"4595d969-33b8-4b17-9ee3-b6c27626823a","year":1999},{"cited_arxiv_id":"","doi":"10.4064/aa-95-3-261-288","is_internal_anchor":false,"ref_index":5,"title":"Y. F. Bilu and R. F. Tichy,The Diophantine equationf(x) =g(y), Acta Arith.95 (2000), no. 3, 261–288, DOI 10.4064/aa-95-3-261-288","work_id":"34a5f9ca-a709-414a-a408-2731a1bd072f","year":2000}],"snapshot_sha256":"c17f69d14a73e8f171fdb3939ec77615523c0f28adf69430c55ff16798509fed"},"source":{"id":"2605.12903","kind":"arxiv","version":1},"verdict":{"created_at":"2026-05-14T18:52:06.488609Z","id":"44377ef2-a7eb-40ea-ae6e-cdcfc2029789","model_set":{"reader":"grok-4.3"},"one_line_summary":"Rational components of f(X)-g(Y)=0 with one geometric point at infinity and projection degree d contribute asymptotically B^{[k:Q]/d} (log B)^{|S|-1} when S-active, while others contribute at most polylogarithmically or finitely many terms.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Rational components with one geometric point at infinity contribute sharp power-log order B^{[k:Q]/d_X(C)} (log B)^{|S|-1} when S-active.","strongest_claim":"For a rational component of f(X)-g(Y)=0 with one geometric point at infinity and projection degree d_X(C) to the X-line, the corresponding contribution has the sharp power-log order B^{[k:Q]/d_X(C)}(log B)^{q_{k,S}}, where q_{k,S}=rk O_{k,S}^* = |S|-1, precisely when its X-parametrization is S-active.","weakest_assumption":"The analysis assumes that after removing the graph components Y=h(X) with f=g o h, the remaining rational components of the curve f(X)-g(Y)=0 can be classified by their number of geometric points at infinity and that the S-activity of the X-parametrization is well-defined and detectable."}},"verdict_id":"44377ef2-a7eb-40ea-ae6e-cdcfc2029789"}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:06c22793829e3365d36dba94f93618c55e13df1cd9678585bb71228e5aad9a02","target":"record","created_at":"2026-05-18T03:09:10Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"ed76e1982b3003ba17d7721bdca813c4e3916995d931b1c37684c06cfd049b8c","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-05-13T02:28:01Z","title_canon_sha256":"a9480d13dce943905f9a7b0f18d5ce84e3ec4845c00316fddf5933f42a9bf6ea"},"schema_version":"1.0","source":{"id":"2605.12903","kind":"arxiv","version":1}},"canonical_sha256":"d94cadf9895391647fc1edb757c0afb76d3f8012a832c6e8bd7e1664e4f9d0c1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d94cadf9895391647fc1edb757c0afb76d3f8012a832c6e8bd7e1664e4f9d0c1","first_computed_at":"2026-05-18T03:09:10.676671Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:09:10.676671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"af7COPI8daQ9bHsZitmvA2r33fkjbzwmjWc0nr5PEnKPigXu+f2HF+QkBHcaCqIYnb0MnpVr97CuuqKuElFIDg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:09:10.677455Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.12903","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:06c22793829e3365d36dba94f93618c55e13df1cd9678585bb71228e5aad9a02","sha256:b0632d051a60f40925020127bc3ed872aa71020efc14f9306ad13926cae254ff"],"state_sha256":"4d209ad42fa65fbdcd1e404147434561d405a8dc0fe303dccdd542073218bf58"}